Applied Statistics and Probability for Engineers

5-7 LINEAR COMBINATIONS OF RANDOM VARIABLES 183

The conclusion for is obtained as follows. Using Equation 5-39, with and

V(Xi)^2 , yields

Another useful result concerning linear combinations of random variables is a reproduc-

tive propertythat holds for independent, normal random variables.

p terms

V 1 X 2  11 p 22 ^2 p 11 p 22 ^2 ^2 p

V 1 X 2 ci 1 p

If with E(Xi)for i1, 2, , p

(5-40a)

if X 1 , X 2 ,, Xpare also independent with V(Xi)^2 for i1, 2, , p,

V 1 X 2  (5-40b)

^2

p

p p

E 1 X 2 

X 1 X 1 X 2 pXp 2 p p

Mean and

Variance of an

Average

w

If X 1 , X 2 ,, Xpare independent, normal random variables with E(Xi)iand

, for i1, 2, , p,

is a normal random variable with

and

V 1 Y 2 c^21 ^21 c^22  22 pc^2 pp^2 (5-41)

E 1 Y 2 c 1  1 c 2  2 pcpp

Yc 1 X 1 c 2 X 2 pcpXp

V 1 Xi 2 ^2 i p

p

Reproductive

Property of the

Normal

Distribution

The mean and variance of Yfollow from Equations 5-37 and 5-39. The fact that Yhas a nor-

mal distribution can be obtained from moment-generating functions discussed in Section 5-9

in the CD material.

EXAMPLE 5-37 Let the random variables X 1 and X 2 denote the length and width, respectively, of a manufac-

tured part. Assume that X 1 is normal with E(X 1 )2 centimeters and standard deviation

0.1 centimeter and that X 2 is normal with E(X 2 )5 centimeters and standard deviation 0.2

centimeter. Also, assume that X 1 and X 2 are independent. Determine the probability that the

perimeter exceeds 14.5 centimeters.

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